We shall give a set $L_1$ of points in three-dimensional space in which a given point $x\in L_1$ has the greatest possible out-degree, hence proving that the out degree for any set of points in three-dimensional space is less or equal than the out degree for our point $x$. To accomplish this, we are going to build the cloud point by point, thus finding the out-degree for HSP-3.

 The next point we choose, $y_1$, must be farther away from $x$ than $y_0$, but we can choose it such that $d(x,y_1)<1+\delta$, for any $\delta>0$. So from now on, we will assume that the points we choose are on the sphere with center $x$ and radius $1$, plus a small $\delta$ for each point.